Crystal structures

Crystal Material, Non-Crystalline Material, Crystal Structure, Space Lattice, Unit Cell, Crystal Systems, and Bravais Lattices, Simple Cubic Lattice, Body-Centered Cubic Structure, Face centered cubic structure, No of Atoms per Unit Cell, Atomic Radius, Atomic Packing Factor, Coordination Number, Crystal Defects, Point Defects, Line Defects, Planar Defects, Volume Defects.

1. Crystal Structure By Ratnadeepsinh Jadeja

2. Crystalline Material Atoms or molecules are arranged in a very regular and orderly fashion in 3D pattern. Strength of these materials are comparatively high. Examples are Metals and Alloys.

3. Non-Crystalline Material Atoms or molecules are arranged in a random manner. Strength of these materials are comparatively lower than crystalline material. Examples are Glass, Wood, Plastics, Rubbers, etc.

4. Crystal Structure Crystal structure is defined as regular repetition of 3D pattern of atoms in a space. A crystalline solid can either be a single crystal or an aggregate of many crystals well separated by well-defined grain boundaries. Solids with many crystals are known as poly-crystalline solids.

5. Space Lattice Lattice means an array of points repeating periodically in 3D. The points of intersection of the lines are called lattice points Space lattice means network of imaginary lines drawn in space such that space is divided into parts of equal size (volume) Combing the above two concepts we say that Atoms are located at the lattice points.

6. Unit Cell Atoms are located upon the points of a lattice. The atoms of forming identical tiny boxes, called unit cells, that fill the volume of the material. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. Unit Cell of Simple Cubic Structure

7. Crystal Systems & Bravais Lattices The crystals are divided into 7 systems based on Axial lengths Interfacial angles Directions of axis of symmetry There are 14 possible ways of arranging points in space lattice so that all the lattice points have exactly same surroundings.

8. Crystal Systems & Bravais Lattices Crystal system Length of Base vectors Angles between axes Bravais Lattices (1) Cubic a = b = c α = β = γ = 90 Cubic primitive / Simple Cubic Body centered cubic Face centered cubic (2) Tetragonal a= b ≠ c α = β = γ = 90 Tetragonal primitive Body centered tetra. (3) Hexagonal a= b ≠ c α = β = 90 , γ = 120 Hexagonal primitive

9. Crystal Systems & Bravais Lattices (4) Rhombohedral a = b = c α = β = γ ≠ 90 Rhombohedral (5) Orthorhombic a ≠ b ≠ c α = β = γ = 90 Orthorhombic primitive Body centered ortho. Orthorhombic Face Base Face Centered Centered (6) Monoclinic a ≠ b ≠ c α = β = 90 ≠ γ Monoclinic primitive Face centered mono. (7) Triclinic a ≠ b ≠ c α ≠ β ≠ γ ≠ 90 Triclinic

10. Most found crystal structures in pure metals are: Simple Cubic Structure Body Centered Cubic Structure (BCC) Face Centered Cubic Structure (FCC) Hexagonal Closed Packed Structure (HCP) Simple Cubic Structure Crystal Systems & Bravais Lattices

11. Simple Cubic Lattice: An SC lattice has 8 corner atoms. Each corner atom is shared among 8 adjoining cubes. Thus, the share of one corner atom to one-unit cell = 1/8 x 8 = 1 Atom No. of Atoms per Unit Cell

12. BCC Lattice: Contribution from corner atoms to one-unit cell= 1/8 x 8 = 1 Contribution from body centered atom = 1 Total contribution = 1+1= 2 atoms No. of Atoms per Unit Cell

13. FCC Lattice: In addition to eight corner atoms, there are six face centered atoms at six planes of the cubes. Each of these six face centered atoms is equally shared by two adjacent cubes. So, Contribution from corner atoms to one-unit cell= 1/8 x 8 = 1 Contribution from face centered atoms to one-unit cell = 1/2 x 6 = 3 Total = 1+3 = 4 atoms No. of Atoms per Unit Cell

14. HCP Lattice: 12 atoms at the corners => 1/6 x 12 = 2 02 face Centered atoms => 1/2 x 02 = 1 03 middle layer atoms => 1 x 03 = 3 Total = 6 atoms No. of Atoms per Unit Cell

15. It is possible to evaluate the radii of atoms by assuming that atoms are spheres in contact with each other in a crystal. The atomic radius r is defined as half the distance between the nearest neighbors in a crystal of a pure element. Atomic Radius

16. Simple Cubic Lattice: One atom is at each of the corners of a cube. If ‘a’ is the lattice parameter i.e. length of the cube edge and ‘r’ is the atomic radius, then a = 2r So, r = a/2 Atomic Radius a

17. Body Centered Cubic Structure: Atomic Radius           For Δ ABF For Δ AFH Atomic Radius of Body Centered Cubic Structure r

18. Face Centered Cubic Structure: Atomic Radius a For Δ ABC         Atomic Radius of Face Centered Cubic Structure r   Or

19. In crystallography, atomic packing factor (APF) or packing fraction is the fraction of volume in a crystal structure that is occupied by atoms. It is dimensionless and always less than unity. For practical purposes, the APF of a crystal structure is determined by assuming that atoms are rigid spheres. The radius of the atoms (spheres) is taken to be the maximal value such that the atoms do not overlap For one-component crystals (those that contain only one type of atom), the APF is represented mathematically by Natoms is the number of atoms in the unit cell Vatom is the volume of an atom Vunit cell is the volume occupied by the unit cell. Atomic Packing Factor

20. Simple Cubic Lattice: Atomic Packing Factor No of Atom in = 1 Atomic Radius =     Volume of Atom = Volume of unit cell =   Atomic Packing Factor = No. of Atom × Volume of Atom Volume of unit cell   = = 0.52

21. Body Centered Cubic Structure: Atomic Packing Factor No of Atom = 2 Volume of Atom = Atomic Radius = Volume of Unit Cell = Atomic Packing Factor = No. of Atom × Volume of Atom Volume of unit cell   = = 0.68      

22. Face Centered Cubic Structure: Atomic Packing Factor No of Atom = 4 Volume of Atom = Atomic Radius = Volume of Unit Cell = Atomic Packing Factor = No. of Atom × Volume of Atom Volume of unit cell   = = 0.74      

23. Hexagonal Cubic Structure: Atomic Packing Factor No of Atom = 6 Volume of Atom = Atomic Radius = Volume of Unit Cell = 3 Sin 60 × c   Atomic Packing Factor = No. of Atom × Volume of Atom Volume of unit cell   = = 0.74     Taking c/a ratio for HCP = 1.633

24. Co-ordination Number defined as the number of nearest atoms directly surrounding a given atom. It can also be defined as the nearest neighbors to an atom in a crystal. Remark : When co-ordination number is larger, the structure is more closely packed. Co-ordination Number

25. Simple Cubic Lattice: Co-ordination Number Consider any atom at a corner of a unit cell. Any corner atom has four nearest neighbor atoms in the same plane and two nearest neighbors in vertical plane (one exactly above and the other exactly below). So, Co-ordination no. for simple cubic structure is 4+2 = 6 The distance between the atom and any of the nearest atom will be equal to lattice parameter i.e. a.

26. Body Centered Cubic Structure: Co-ordination Number Consider any atom at a corner of a unit cell. Any corner atom has one body centered atom close to it as shown below. So, Co-ordination no. for BCC structure is 8 The distance between the atom and any of the nearest atom is √3a/2; where a is the lattice parameter.

27. Face Centered Cubic Structure: Co-ordination Number Consider any atom at a corner of a unit cell. For any corner atom, there will be 4 face centered atoms of the surrounding unit cells in its own plane, 4 face centered atoms below its plane and 4 face centered atoms above its plane. So, Co-ordination no. for FCC structure is 4+4+4 = 12 The distance between the atom and any of the nearest atom is a/√2; where a is the lattice parameter.

28. Hexagonal Cubic Structure: Co-ordination Number In HCP, each atom in one layer is located directly above or below interstices among three atoms in the adjacent layers. So, each atom touches three atoms in the layer below its plane, six atoms in its own plane, and three atoms in the layer above. So, Co-ordination no. for HCP structure is 3+6+3 = 12

29. Sr. No. Crystal Structure No of Atom Atomic Radius Atomic Packing Factor Co-ordination Number 1 SC 1 0.52 6 2 BCC 2 0.68 8 3 FCC 4 0.74 12 4 HCP 6 0.74 12        

30. A perfect crystal, with every atom of the same type in the correct position, does not exist. All crystals have some defects. Defects contribute to the mechanical properties of metals. Metals are not perfect neither at the macrolevel and nor at the microlevel Crystal Defects

31. There are basic classes of crystal defects: P oint defects, which are places where an atom is missing or irregularly placed in the lattice structure. Point defects include lattice vacancies, self-interstitial atoms, substitution impurity atoms, and interstitial impurity atoms L inear defects, which are groups of atoms in irregular positions. Linear defects are commonly called dislocations. Planar defects, which are interfaces between homogeneous regions of the material. Planar defects include grain boundaries, stacking faults and external surfaces. Volume defects, the absence of several atoms to form internal surfaces in the crystal. They have similar properties to microcracks because of the broken bonds at the surface. Crystal Defects

32. A Point Defect involves a single atom change to the normal crystal array. There are three major types of point defect: Vacancies, Interstitials and Impurities. They may be built-in with the original crystal growth or activated by heat. They may be the result of radiation, or electric current etc , etc. Point Defects A Vacancy is the absence of an atom from a site normally occupied in the lattice. An Interstitial is an atom on a non-lattice site. There needs to be enough room for it, so this type of defect occurs in open covalent structures, or metallic structures with large atoms. Vacancy Point Defect Interstitial Point Defect

33. Point Defects The defect forms when an atom or smaller ion leaves its place in the lattice, creating a vacancy, and becomes an interstitial by lodging in a nearby location. Frenkel Point Defect This defect forms when oppositely charged ions leave their lattice sites and become incorporated for instance at the surface, creating oppositely charged vacancies. These vacancies are formed in stoichiometric units, to maintain an overall neutral charge in the ionic solid. Schottky Point Defect

34. Line Defects Edge Dislocation Screw Dislocation An  Edge dislocation  in a Metal may be regarded as the insertion (or removal) of an extra half plane of atoms in the crystal structure. A   Screw Dislocation  changes the character of the atom planes. The atom planes no longer exist separately from each other. They form a single surface, like a screw thread, which "spirals" from one end of the crystal to the other. (It is a  helical  structure because it winds up in 3D, not like a spiral that is flat.)

35. Planar Defects Grain Boundaries Tilt Boundaries A Grain Boundary is a general planar defect that separates regions of different crystalline orientation within a polycrystalline solid. The atoms in the grain boundary will not be in perfect crystalline arrangement. Grain boundaries are usually the result of uneven growth when the solid is crystallizing. A Tilt Boundary, between two slightly mis-aligned grains appears as an array of edge dislocations.

36. Planar Defects Twin Boundaries A Twin Boundary happens when the crystals on either side of a plane are mirror images of each other. The boundary between the twinned crystals will be a single plane of atoms. There is no region of disorder and the boundary atoms can be viewed as belonging to the crystal structures of both twins. Twins are either grown-in during crystallization, or the result of mechanical or thermal work.

37. Volume Defects Volume defects are  Voids , i.e. the absence of several atoms to form internal surfaces in the crystal. They have similar properties to microcracks because of the broken bonds at the surface.

38. Thank You